Wavelength using a diffraction grating

Practical method

Aim

Determine the wavelength of laser light using a diffraction grating and several orders of maxima.

Variables

Independent variable: order n
Dependent variable: diffraction angle θ
Control variables: the same grating and laser; normal incidence.

Apparatus

• Laser
• Diffraction grating (300 lines per mm)
• Screen
• Metre rule and tape measure
• Stand for the grating

Method

1. Shine the laser normally through the grating onto a screen about 1 m away; measure the grating–screen distance D.
2. Measure the distance between matching left and right maxima for each order and halve it to find x.
3. Calculate θ from tan θ = \(\dfrac{x}{D}\).
4. Calculate d = \(\dfrac{1}{300 \times 10^{3}}\) m and λ = d \(\dfrac{\sin \theta }{n}\) for each order.
5. Plot sin θ against n: gradient = \(\dfrac{\lambda }{d}\).

Sample results

300 lines per mm, so d = 3.33 × 10⁻⁶ m; D = 1.000 m.

Analysis

sin θ against n is a straight line through the origin with gradient 0.195. λ = d × gradient = 3.33 × 10⁻⁶ × 0.195 = 6.50 × 10⁻⁷ m = 650 nm.

Uncertainty

x is measured to ±2 mm; for the third order (0.721 m) this is 0.3%. Higher orders give smaller percentage uncertainty, which is why several orders and a graph are used.

Errors and improvements

Risk assessment

Common mistakes

• Using lines per mm directly as d.
• Using sin θ ≈ \(\dfrac{x}{D}\) at large angles.
• Forgetting to halve the left–right distance.

Worked analysis

Second order: x = 0.424 m at D = 1.000 m, so θ = tan⁻¹(0.424) = 23.0°. λ = d \(\dfrac{\sin \theta }{2}\) = \(\dfrac{3.33 \times 10^{-6} \times 0.391}{2}\) = 6.51 × 10⁻⁷ m.

Common misconception

The grating spacing d is the distance between lines (1 ÷ lines per metre), not the number of lines per millimetre.